A study of consumer smoking habits includes 198 people in the 18-22 age bracket (40 of whom smoke), 125 people in the 23-30 age bracket (31 of whom smoke), and 90 people in the 31-40 age bracket (30 of whom smoke). If one person is randomly selected from this sample, find the probability of getting someone who is age 23-30 or smokes

To solve this, we need to determine the probability of selecting someone who is either in the age range of 23-30 or smokes. This is an example of finding the probability of the union of two events.

Let:
- \( A \) be the event that a person is in the 23-30 age bracket.
- \( B \) be the event that a person smokes.

We are tasked with finding \( P(A \cup B) \), the probability that a randomly selected person is either in the 23-30 age bracket or smokes.

The formula for the union of two events is:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]
Where:
- \( P(A) \) is the probability that a person is in the 23-30 age bracket.
- \( P(B) \) is the probability that a person smokes.
- \( P(A \cap B) \) is the probability that a person is both in the 23-30 age bracket and smokes.

### Step 1: Find the total number of people
The total number of people is:
\[
198 \, \text{(18-22)} + 125 \, \text{(23-30)} + 90 \, \text{(31-40)} = 413
\]

### Step 2: Calculate \( P(A) \)
The number of people in the 23-30 age bracket is 125. So the probability \( P(A) \) is:
\[
P(A) = \frac{125}{413}
\]

### Step 3: Calculate \( P(B) \)
The number of smokers is \( 40 \, (\text{18-22}) + 31 \, (\text{23-30}) + 30 \, (\text{31-40}) = 101 \). So the probability \( P(B) \) is:
\[
P(B) = \frac{101}{413}
\]

### Step 4: Calculate \( P(A \cap B) \)
The number of people who are both in the 23-30 age bracket and smoke is 31. So the probability \( P(A \cap B) \) is:
\[
P(A \cap B) = \frac{31}{413}
\]

### Step 5: Use the formula
Now we apply the formula for the union of two events:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]
Substitute the values:
\[
P(A \cup B) = \frac{125}{413} + \frac{101}{413} - \frac{31}{413} = \frac{195}{413}
\]

Thus, the probability of selecting someone who is either in the 23-30 age bracket or smokes is:
\[
P(A \cup B) = \frac{195}{413} \approx 0.472
\]

So, the probability is approximately **0.472** or **47.2%**.

Would you like further clarification or details? Here are 5 related questions:

1. What is the probability of selecting someone who is in the 18-22 age bracket and smokes?
2. How would the probability change if the number of smokers in the 31-40 age bracket increased by 5?
3. Can you calculate the probability of selecting a non-smoker from the 23-30 age bracket?
4. How would you compute the probability of selecting a smoker from the entire sample?
5. How does the complement rule apply to this problem?

**Tip:** When dealing with probability questions involving "or," remember to subtract the overlap (i.e., the intersection) to avoid double-counting.

A study of consumer smoking habits includes 198 people in the 18-22 age bracket (40 of whom smoke), 125 people in the 23-30 age bracket (31 of whom smoke), and 90 people in the 31-40 age bracket (30 of whom smoke). If one person is randomly selected from this sample, find the probability of getting someone who is age 23-30 or smokes.

Find the probability of selecting someone who is either in the 23-30 age bracket or smokes from a study of 413 people with detailed calculations. Learn about the union of events, including key probability concepts and formulas, suitable for high school students.

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